Admissible metrics on compact K\"ahler varieties

TL;DR

This paper explores the existence of admissible Hermitian and Hermitian-Yang-Mills metrics on coherent sheaves over compact Kähler varieties, linking metric existence to Sobolev inequalities on singular spaces.

Contribution

It establishes conditions under which admissible Hermitian metrics exist on reflexive sheaves, particularly for slope stable cases, advancing the understanding of metric structures on singular Kähler varieties.

Findings

Existence of admissible Hermitian metrics on reflexive sheaves is linked to Sobolev inequalities.

For slope stable sheaves, admissible Hermitian-Yang-Mills metrics exist under certain conditions.

The results depend on proving uniform Sobolev inequalities on singular spaces.

Abstract

Let $X$ be a normal compact K\"ahler variety, and $\mathcal{F}$ a coherent reflexive sheaf on $X$. We investigate the existence of admissible Hermitian metrics on $\mathcal{F}$. If moreover $\mathcal{F}$ is slope stable, we also study the existence of admissible Hermitian-Yang-Mills metrics on it. The existence will hold if one can prove a uniform Sobolev inequality on singular spaces.